Measurement of Dynamic Portfolio VaR Based on Mixed Vine Copula Model

Transcription

1 Journal of Fnance and Accountng 207; 5(2): do: 0.648/j.jfa ISSN: (Prnt); ISSN: (Onlne) Measurement of Dynamc Portfolo VaR Based on Mxed Vne Copula Model Zhao Ru-bo, Tan Y-xang, Tan We, Chen Xu-rong School of Management and Economcs, Unversty of Electronc Scence and Technology of Chna, Chengdu, Chna Emal address: (Zhao Ru-bo), (Tan Y-xang), (Tan We), (Chen Xu-rong) To cte ths artcle: Zhao Ru-bo, Tan Y-xang, Tan We, Chen Xu-rong. Measurement of Dynamc Portfolo VaR Based on Mxed Vne Copula Model. Journal of Fnance and Accountng. Vol. 5, No. 2, 207, pp do: 0.648/j.jfa Receved: March 7, 207; Accepted: Aprl 4, 207; Publshed: Aprl 2, 207 Abstract: The measurement of portfolo VaR has been a hot ssue n the feld of the academc and the ndustry. Ths paper apples three knds of Vne Copula model to descrbe hgh-dmensonal dependency structure between multple assets, ntroduces mxed bnary copula functon to mprove the accuracy of tal dependence structure. We use sx mportant stock markets as stock portfolo to test ths model. The emprcal results show that ntroducng mxed Copula functon can mprove the measurement relablty of Vne Copula model, and the relablty of mxed R-Vne model s hghest n three knds of mxed Vne Copula models. Keywords: Mxed Copula, Vne Copula, Dynamc of VaR, Portfolo. Introducton Fnancal rsk management has always been a hot and dffcult ssue for nvestors, government fnancal management and fnancal academa []. Especally n recent years, the outbreaks of global rsk crss hghlght the mportance of fnancal rsk management and urgency. In fnancal rsk management, the most crtcal job s how to accurately measure fnancal rsks. Only measurng fnancal rsks accurately can we ensure the effectveness of rsk management. At present, the manstream approach n a sngle asset or portfolo rsk measure s VaR (Value at Rsk) [2]. Measurng the VaR of a sngle asset s smpler, but because the fnancal nsttuton and the nvestor tend to nvest n the form of a portfolo, and to measure the VaR of the portfolo, t must frst depct the hgh-dmensonal structure of the portfolo. Therefore, how to accurately characterze the nterdependence between the portfolo wll be the key to measurng the portfolo VaR. Many scholars have shown that fnancal assets have many typcal characterstcs such as spkes and fat tal, non-normalty, volatlty aggregaton and leverage. At the same tme, the dependent structure between fnancal assets s non-lnear [3]. Therefore, the lnear correlaton model, Granger causalty test and other lnear model cannot accurately descrbe the non-lnear dependent structure of fnancal assets [4]. And then based on the above model s dffcult to accurately measure the portfolo VaR. Copula functon effectvely overcomes the shortcomngs of the above lnear model, not only can descrbe the nonlnear dependency structure between fnancal assets, but also can nclude typcal fact characterstcs nto the study constrants. Snce then, Copula functon has been wdely used by scholars n the felds of fnancal rsk management and rsk measurement, and has acheved many mportant practcal achevements. At present, Scholars use bnary copula functons n the fnancal research. When the number of varables s more than two, the bnary copula functon wll face dmenson curse. Although the multdmensonal Copula functon s used to solve the problem of hgh-dmensonal dependency structure, the multvarate Copula functon assumes that the dependent structure between dfferent fnancal assets s same, and ths s not consstent wth the actual stuaton, so that the multvarate Copula functon cannot accurately depct the nterdependent structure between multple assets [5]. Recently, the Vne Copula model has effectvely solved the problem of hgh-dmensonal dependency structure of multple fnancal assets, and vne copula functon has more

2 8 Zhao Ru-bo et al.: Measurement of Dynamc Portfolo VaR Based on Mxed Vne Copula Model flexblty than multvarate Copula functon. Joe, Bedford and Cooke have made groundbreakng work for the constructon of the Vne Copula model [6] [7] [8]. Vne copula model s layered to depct the dependent structure, and the model greatly reduce the dffculty of characterzng hgh-dmensonal dependent structure. In addton, the Vne Copula model can be used to select the approprate bnary Copula functon accordng to the dfferent characterstcs of the object-dependent structure, so that the Vne Copula model has more flexblty and practcablty n characterzng the hgh-dmensonal dependency structure. Bedford and Cooke proposed R-Vne, R-Vne Copula s more flexble and complex, the dffculty of estmaton s relatvely large. The applcaton s far less than ts two specal crcumstances: C-Vne and D-Vne [8] [9]. Therefore, to hghlght the flexblty and advantage of R-Vne model n characterzng the structure of hgh-dmensonal dependences, ths paper not only uses C-Vne and D-Vne, but also uses R-Vne. In the use of vne copula model to descrbe hgh-dmensonal dependent structure, they chose bnary copula functon based on the AIC and other ndcators. Although a relatvely optmal Vne Copula model can be selected, ths wll ncrease the rsk of Vne Copula's model. In addton, a sngle bnary Copula functon s usually only able to descrbe some of the tal dependent structure or cannot descrbe the tal dependent structure, and tal dependent structure represents the degree of fnancal assets n extreme cases, the tal-dependent structure s of great mportance to accurately measure the extreme fnancal rsks of fnancal assets. Clayton, Frank and Gumbel can characterze the lower end dependent structure, symmetrc dependent structure and upper tal dependent structure [0]. To make full use of the characterstcs of Copula functon, ths paper construct a mxed copula functon to descrbe the dependent structure of two-dmensonal varables n the vne Copula model. Based on the above analyss and understandng, ths paper wll take the world's sx major stock markets as the study object, usng the ARMA-GJR-t model to capture the mportant typcal factual characterstcs of the yeld and volatlty of fnancal assets; amng at the complex hgh-dmensonal dependency structure between the portfolo, three Vne Copula models are ntroduced, At the same tme, the mxed Copula functon s used n the Vne Copula model to descrbe the asymmetrc tal dependent structure between the bnary varables, n order to carry out the VaR dynamc rsk measure; And then use the rgorous lkelhood rato test to test the relablty of the stock market portfolo rsk model. And tres to prove the followng two conclusons through the relevant theoretcal analyss and emprcal research: The R-Vne Copula model s more relable than the C-Vne and D-Vne Copula models n the rsk measure. The hybrd Copula functon can mprove the relablty of the Vne Copula model for the dynamc rsk measure. So far, many scholars have used a varety of methods to carry out a lot of research on portfolo rsk measurement, and acheved frutful research results. Alou et al. used the bnary Copula functon to study the fnancal rsk of the fnancal crss between the BRIC countres and the US stock market, and measured the rsk of the bnary portfolo []. Deng et al. used the Par Copula model to measure portfolo rsk, whle also usng Mean-CVaR to optmze the portfolo [2]. Zhou et al used multple Copula functon to measure the rsk of multple fnancal portfolo [3]. Brechmann et al. used the R-Vne Copula model to predct the portfolo rsk value. The results show that the R-Vne model measures better than the DCC model [4]. Weß et al. used the Vne Copula model to predct the portfolo rsk. The results show that the Vne Copula model can accurately measure the rsk of portfolo [5]. Fan Guobn et al. used the C-Vne Copula model to characterze the nonlnear dependences between multple fnancal assets. The emprcal results show that the C-Vne Copula model can descrbe the complex tal-dependent structure of fnancal assets more accurately than the tradtonal mult-copula functon [6]. Gao used the vne Copula model to measure the rsk of multple assets [7]. Zhang et al. used the R-Vne, C-Vne and D-Vne models to explore the nterdependent structure of dfferent fnancal assets n Chna. The emprcal results show that Chna's fnancal market shows thck-taled correlaton and asymmetry [8]. Ma Feng used the Vne Copula model to carry out dynamc rsk forecastng of the stock market portfolo [9]. Zhao Lu used the multvarate t-copula functon to measure the rsk of the energy portfolo and optmze the weght of the portfolo [20]. Although these scholars have acheved satsfactory results n the relevant studes, these studes stll have the followng two shortcomngs: these scholars use only multple elements to descrbe the same dependent structure of the Copula functon, but between fnancal assets Dependent structure may not be completely consstent, then the use of multple Copula functon wll be questonable; Although there are a few scholars usng Vne Copula functon to descrbe hgh-dmensonal dependent structure, but they used bnary copula functon n vne copula model, and these bnary Copula functon cannot descrbe the asymmetrc tal dependent structure. Ths artcle not only uses C-Vne, D-Vne, but also uses more flexble and complex R-Vne n characterzng the portfolo structure. In addton, the hybrd Copula functon s ntroduced nto the Vne Copula model, and a new mxed Vne Copula model s constructed. Therefore, compared wth other scholars, the hybrd vne copula can measure the dependent structure more accurate especally n the fnancal crss, whch has a promnent advantage n measurng the dynamc rsk of the portfolo. The logcal structure of ths paper s roughly arranged as follows, the followng s the second part of ths artcle, constructng the dynamc VaR measure model based on the mxed vne copula model. The thrd part uses the constructed rsk forecastng model to carry on the dynamc rsk measure to the portfolo, the last part s the concluson.

3 Journal of Fnance and Accountng 207; 5(2): Constructon of Dynamc Rsk Measurement Model of Portfolo 2.. Vne Copula Model Sklar theorem suggests that the d-dmensonal cumulatve dstrbuton functon F can be decomposed nto an edge dstrbuton functon and a Copula Functon C, whch can be expressed as [2]: F( X, X,, X ) = C( F ( x ),, F ( x ) () 2 d d d The correspondng jont densty functon can be expressed as: (,,, ) = ( ( ),, ( )) d f x x x c F x F x f ( x ) (2) 2 d d d = Where c s the Copula functon densty functon and f s the edge densty functon. As the number of varables ncreases, t s ncreasngly complcated to measure the jont dstrbuton between these varables. Joe, Bedford and Cooke proposed the Vne Copula model, the hgh-dmensonal jont densty functon s decomposed nto several bnary jont dstrbuton densty functons and correspondng edge densty functons, so that the jont dstrbuton of multvarate varables s smplfed. there are a varety of logcal structure n the vne copula model decomposton, Bedford et al. ntroduced the vne to descrbe these decomposton structure. The commonly used decomposton structures nclude Canoncal Vne (C-Vne), Drawable Vne (D-Vne) and Regular Vne (R-Vne) [22]. The structure of C-Vne and D-Vne s more fxed and smple than R-Vne, and the structure of R-Vne s more flexble. A d-dmensonal C-Vne or D-Vne decomposton structure can be represented by a d- tree, where the j-th tree has d + -j nodes, d-j edges, and each edge corresponds to a bnary Copula functon. The edge of the treet j becomes a node n the treet j +. For the d-dmensonal jont densty functon can be decomposed nto the d( d ) / 2 edge and the correspondng edge densty functon accordng to C-Vne and D-Vne. The C-Vne and D-Vne densty functons can be expressed as follows: d d = d j d k k,, j j j + j k = j = = f ( x,, x ) f ( x ) c ( F( x x,, x ), F( x x,, x )) (3) d d = d j d k k, +,, + j j j j j j j k = j= = f ( x,, x ) f ( x ) c ( F( x x,, x ), F( x x,, x )) (4) Where s the herarchy of the tree, and j s the edge of each tree. Compared wth C-Vne and D-Vne, the decomposton structure of R-Vne s not fxed, So R-vne s relatvely more flexble. The densty functon of R-Vne s: d d d f ( x,, x ) f ( x ) c ( F, F ) (5) = d k k m, m, m +, m, m, m +, m, m, m +, m k k k k n k k k k n k k k n, k,,,,,,, k = k = = k Mxed Vne Copula Model Constructon and Parameter Estmaton When usng the Vne Copula model studes the d-dmensonal fnancal asset-dependent structure, except that the approprate decomposton structure s selected, t s also necessary to further determne the bnary Copula functon n the decomposton structure. At present, n the relevant research, scholars often n terms of AIC and BIC. The bnary Copula functons n the decomposed structure are selected one by one from the bnary Copula functons such as Gauss and t-copula. Gumbel, Clayton and Frank consttute a hybrd Copula functon just to accurately measure the asymmetrc tal dependent structure, Therefore, ths paper wll use the hybrd Copula functon n the Vne Copula model to descrbe the bnary varable dependent structure. The mxed Copula functon dstrbuton functon s shown below: C( u, v; Ψ ) = π C ( u, v; θ ) + π C ( u, v; λ) + C ( u, v; α ) (6) c 2 f g where Cc s Clayton functon, C f s rank functon, C g s Gumbel functon, Ψ s the parameter to be estmated, ω, =,2,3 s weght of three Copula functons, 3 = ω =. ths paper wll use the maxmum lkelhood parameter estmaton method to estmate the mxed Vne Copula model parameters, the specfc estmaton process s as follows: ) To sort the nodes n the decomposton structure. For the C-Vne decomposton structure, ths paper determnes the order of the nodes based on the sum of the Kendall coeffcents of each tree. When the order of the nodes n the frst layer tree s determned, the nodes n the remanng trees wll be fxed. In ths paper, the D-Vne node orderng wll be determned based on Brechmann s method [22]. R-Vne s more flexble than C-Vne and D-Vne, and t s necessary to frst fnd the optmal R-Vne matrx (RVM) to determne the order of the nodes. In ths paper, we refer to the "maxmum spannng tree" method proposed by Brecchmann to select the

4 83 Zhao Ru-bo et al.: Measurement of Dynamc Portfolo VaR Based on Mxed Vne Copula Model approprate RVM, and then determne the arrangement of nodes n R-Vne decomposton based on RVM content. 2) Estmate the parameters of each edge n tree j. Based on the decomposed structure that has been determned, the parameters of the bnary mxed Copula functon represented by each edge are estmated usng the maxmum lkelhood estmaton method, and the correspondng condtonal dstrbuton functon. R-Vne model condtonal dstrbuton functon needs to be determned accordng to the decomposton structure. 3) The Copula functon parameters for each edge on tree j+ are estmated based on the calculated condtonal dstrbuton functon untl the Copula functon parameters are mxed for each edge of all trees Margnal Dstrbuton Model As the stock market returns often have fat tal, non-normal, autocorrelaton, volatlty asymmetry and other characterstcs, Therefore, ths paper chooses the student t dstrbuton to descrbe the dstrbuton characterstcs of the stock market returns. The ARMA (, ) -GJR (, ) -t model s used to characterze the autocorrelaton, heteroskedastcty and fluctuaton asymmetry of stock market returns. margnal dstrbuton model s as follows: R = u + φ R + θ ε + ε (7) nt n n n, t n n, t nt ε = σ ξ (8) nt nt nt σ = ω + β σ + α ε + γ ε I( ε < 0) (9) nt n n n, t n n, t n n, t n, t 2 Where σ nt s condtonal varance, ξ nt s standard resdual and obey the standard student t dstrbuton. I ( ) s the nstructon functon. When the condton s satsfed, the value s, otherwse t s Portfolo Dynamc VaR Measure At present, the rsk management manly uses VaR (Value at Rsk) to measure, VaR measures the maxmum possble loss of a portfolo at a gven confdence level for a certan perod n the future. The mathematcal expresson s as follows: d t t = P( ω X VaR ( α) Ω ) = α (0) Where d s number of assets, ω s the weght of the -th asset ω =. d = As drect estmate of portfolo VaR s very dffcult, most scholars use Monte Carlo smulaton method. Therefore, ths artcle wll also use Monte Carlo smulaton method, the specfc steps are as follows: () Use ARMA (, ) -GJR (, ) -t to estmate the yeld of each asset n the portfolo, Extractng the volatlty σ, t of each stock market condtons. On ths bass, the standard yeld seres s transformed by probablty ntegral, and the unform dstrbuton sequence of.. d (0, ) s obtaned, expressed as Udata ; (2) Usng the resultng sequence Udata, the nodes of R-Vne, C-Vne and D-Vne are sorted frst. Then the parameters of the three mxed Vne Copula models are estmated by usng the maxmum lkelhood estmaton method. Monte Carlo smulaton based on the estmated model parameters was smulated 5000 tmes per day, and fnally the T dmenson array was obtaned. (3) The smulaton array s nversed based on student t dstrbuton. The dstrbuton of the resdual value ( z,, z ) of the combned assets s obtaned, and then, t 6, t the dstrbuton of the daly returns of the combned assets s obtaned accordng to the formula (7). 6 X = ω ( u + σ z ) () t, t, t, t = (4) Fnally, accordng to Equaton 0, we can get the dynamc VaR value of the portfolo every day durng the research perod. 3. Emprcal Results and Analyss 3.. Sample Selecton In ths paper, the Shangha Composte Index (SSEC), the Hang Seng Index (HSI), the Nkke 225 Index (N225), the Frankfurt DAX Index (DAX), the London Fnancal Tmes 00 Index (FTSE00), the S & P 500 Index (S & P500) are taken as mportant stock market representatves. The study sample s the daly closng prce from January 3, 202 to December 3, 205. The data comes from the Resset database. As each stock market tradng tme s not the same, to mantan the sx stock market yelds data length consstent, ths paper elmnates the nconsstency of each stock tradng data, and fnally the sample number s 7. As can be seen from the descrptve statstcal analyss results n Table, The sx stock market sequences show sgnfcant "spkes" and "fat tal" characterstcs, and do not obey the normal dstrbuton assumptons, The J-B test results also confrm the non-normalty of the stock market ncome sequence. In addton, the ARCH effect test statstc LM ndcates that the sx stock markets have a sgnfcant ARCH effect, that s, the sx stock market sequences have fluctuatng aggregaton effect and fluctuatng tme-varyng characterstcs.

5 Journal of Fnance and Accountng 207; 5(2): Table. The results of descrptve statstcal analyss of log yelds n each stock market. Mean Std Skewness Kurtoss J-B LM(5) LM(0) LM(5) SSEC *** *** *** *** *** *** HSI *** *** *** *** *** *** N *** *** *** *** *** *** DAX ** *** *** *** *** *** FTSE *** *** *** *** *** *** S&P *** *** 299. *** *** *** *** Note: ***, **, respectvely, at %, 5% sgnfcant level of sgnfcance, J-B t s Jarque-Bera statstcs, LM s Engle's ARCH test statstcs (brackets ndcate the lag order) Margnal Dstrbuton Model Estmaton Results Table 2. Parameter of margnal dstrbuton model. SSEC HSI N225 DAX FTSE00 S&P500 u φ θ ω β α γ v Q(5) K-S Note: K-S s Kolmogorov-Smrnov test P value. As there are non-normalty, spkes and fat tal and other characterstcs n the sx stock market returns sequence. In ths paper, the ARMA (, ) -GJR (, ) -t model s used to analyze the margnal dstrbuton of the sx stock markets. The parameters result of the margnal dstrbuton model s shown n Table 2. And then extract the stock market standard rate of return sequence, perform the probablty ntegral transformaton. To test the fttng ablty of the margnal dstrbuton model for the sx stock market return seres, ensurng that the seres after the probablty ntegral transformaton s unformly dstrbuted by.. d (0, ). We also used the Kolmogorov-Smrnov (K-S) test, whch showed that the seres s unformly dstrbuted. In addton, we also used the Ljung-Box Q test, the test results show that the stock market seres does not exst autocorrelaton after the ntegral transformaton. Therefore, the margnal dstrbuton model ARMA (, ) -GJR (, ) -t can descrbe the sx stock market returns seres. On ths bass, the Vne Copula model can be used to analyze the nterdependent structure between the sx stock markets, measure the dynamc rsk of portfolo Portfolo Rsk Measurement Model Relablty Test As ths paper taken the world s sx major stock market as research object, at the same tme ths paper uses mxed Copula functon n the Vne Copula model. Therefore, whether t s C-Vne, D-Vne or R-Vne, the estmated parameters are up to 80, consderng the artcle space constrants, here we no longer gve three types of rattan structure. Fgure. Stock market portfolo VaR.

6 85 Zhao Ru-bo et al.: Measurement of Dynamc Portfolo VaR Based on Mxed Vne Copula Model Fgure shows the VaR measurement results for the stock market portfolo based on the mxed C-Vne, D-Vne and R-Vne models at the 5% quantle level durng the sample perod. It can be seen from Fgure that the three mxed Vne Copula models can accurately measure the dynamc rsk of the stock market portfolo. After measurng the dynamc rsk of the stock market portfolo through dfferent models, an mportant job s to test the relablty of the rsk measurement model. In ths paper, we wll use the condtonal coverage test proposed by Chrstoffersen (998) [23], because the condtonal coverage test consders the falure number and ndependence, so the test has a more comprehensve. To be able to compare the dfferences between the mxed Vne Copula model and the Vne Copula model n the portfolo rsk measure, we wll backtest three Vne Copula models. The backtestng results are shown n table 3 under the same weght condtons. Table 3. Portfolo Rsk Measurements Back-testng. % 5% 0% K(E) cc K(K2) cc K(K2) cc C-mxed 6(7.) (35.55) (7.) D-mxed 6(7.) (35.55) (7.) R-mxed 7(7.) (35.55) (7.) R-Vne 8(7.) (35.55) (7.) C-Vne 0(7.) (35.55) (7.) 0.39 D-Vne 7(7.) (35.55) (7.) Note: K and E represent the actual falure of VaR measure and the number of theoretcal falures, respectvely, for the condtonal coverage test P value, The hgher the P value, the hgher the relablty of the model. It can be seen from Table 3 that the relablty of the three mxed Vne Copula models s hgher than three Vne Copula models at three confdence levels, whch shows that the bnary hybrd Copula functon can more accurately descrbe the nterdependence structure between bnary fnancal assets, especally the extremely dependent structure between fnancal assets. Fnally, the mxed Vne Copula model s more accurate and relable n measurng the dynamc rsk of a portfolo asset. In addton, we can see that the relablty of the mxed R-Vne model s hgher than the mxed C-Vne and D-Vne model, whch shows that the mxed R-Vne model s more accurate n descrbng the dependent structure of multcomponent fnancal assets than mxed C-Vne and mxed D-Vne models. Snce the equal weght s a very specal case. To ensure the robustness of the test results n ths paper, we refer to the Mean-CVaR method to optmze the weght of the portfolo and yeld a new portfolo weght. The new weghts for the portfolo are: , 0.08, 0.02, 0.099, 0.44, And then backtest agan, the result s shown n Table 4. It can be seen from Table 4 that the test results not only prove the relablty of the backtestng result under the same weght condton, but also further prove the superorty of the mxed. To be able to observe the VaR measure of the combned asset clearly, we only report the results of the VaR measure under the condton of 5% quantle and three mxed Vne Copula models. Vne Copula model, especally the mxed R-Vne Copula model n measurng the dynamc rsk of the portfolo assets. Table 4. Portfolo Rsk Measurements Back-testng. % 5% 0% K(K2) cc K(K2) cc K(K2) cc C-mxed 6(7.) (35.55) (7.) 0.39 D-mxed 6(7.) (35.55) (7.) R-mxed 8(7.) (35.55) (7.) R-Vne 0(7.) (35.55) (7.) C-Vne 4(7.) (35.55) (7.) D-Vne 0(7.) (35.55) (7.) Note: K and E represent the actual falure of VaR measure and the number of theoretcal falures, respectvely, for the condtonal coverage test P value, The hgher the P value, the hgher the relablty of the model. 4. Concluson It s a hot topc to study the hgh-dmensonal dependency structure among many fnancal assets, especally the tal-dependent structure of fnancal assets durng the fnancal crss. Therefore, we ntroduce a more flexble and accurate Vne Copula model, and we use the mxed Copula functon composed of Clayton, Frank and Gumbel to descrbe the bnary dependency structure to measure asymmetrc tal dependent structure between fnancal assets accurately. And takng the world s sx major stock markets as research object, usng Monte Carlo smulaton technology, applyng three mxed vne Copula model to measure the dynamc rsk of stock portfolo, carryng out a rgorous Back-testng. The emprcal results show that the relablty of the three mxed Vne Copula models s sgnfcantly better than other three types of Vne Copula models. In addton, the relablty of the mxed R-Vne model s superor to the mxed C-Vne and mxed D-Vne models, whch shows that the mxed R-Vne model can measure the stock portfolo dependence structure accurately. Acknowledgement The author would lke to thank the Natonal Socal Scence Fund of the People s Republc of Chna for fnancally supportng ths study under Contract No. 4BJY74. 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